The electric and magnetic fields along with the electric and
magnetic vector potentials expanded in the rectangular coordinate
system are shown below.
\begin{align}
\vec{H} &= H_x\hat{x}+H_y\hat{y}+H_z\hat{z}\label{eqn:Hexp}\\
\vec{E} &= E_x\hat{x}+E_y\hat{y}+E_z\hat{z}\label{eqn:Eexp}\\
\vec{A} &= A_x\hat{x}+A_y\hat{y}+A_z\hat{z}\label{eqn:Aexp}\\
\vec{F} &= F_x\hat{x}+F_y\hat{y}+F_z\hat{z}\label{eqn:Fexp}
\end{align}
Substituting the expanded vectors (\ref{eqn:Hexp}),
(\ref{eqn:Aexp}) and (\ref{eqn:Fexp}) into
(\ref{eqn:Et3})--(\ref{eqn:Et4}) yields the expanded magnetic
field components.
\begin{align}
    H_x&=\frac{\partial A_z}{\partial y}-\frac{\partial{A_y}}{\partial{z}}+\frac{1}{\sigma_m+j\omega\mu}\left[\biggl(\frac{\partial^2}{\partial x^2}+(k^2=-\gamma^2)\biggr){F_x}+\frac{\partial^2 F_y}{\partial x\partial y}+\frac{\partial^2{F_z}}{\partial x\partial{z}}\right]\label{eqn:Hxt}\\
    H_y&=\frac{\partial A_x}{\partial z}-\frac{\partial{A_z}}{\partial{x}}+\frac{1}{\sigma_m+j\omega\mu}\left[\frac{\partial^2 F_x}{\partial x\partial{y}}+\biggl(\frac{\partial^2}{\partial{y}^2}+(k^2=-\gamma^2)\biggr)F_y+\frac{\partial^2 F_z}{\partial{y}\partial{z}}\right]\label{eqn:Hyt}\\
    H_z&=\frac{\partial A_y}{\partial x}-\frac{\partial{A_x}}{\partial{y}}+\frac{1}{\sigma_m+j\omega\mu}\left[\frac{\partial^2 F_x}{\partial x\partial{z}}+\frac{\partial^2{F_y}}{\partial y\partial{z}}+\biggl(\frac{\partial^2}{\partial{z}^2}+(k^2=-\gamma^2)\biggr)F_z\right]\label{eqn:Hzt}
\end{align}
Substituting the expanded vectors (\ref{eqn:Eexp}),
(\ref{eqn:Aexp}) and (\ref{eqn:Fexp}) into
(\ref{eqn:Et3})--(\ref{eqn:Et4}) yields the expanded electric
field components.
\begin{align}
    E_x&=\frac{\partial{F_y}}{\partial{z}}-\frac{\partial{F_z}}{\partial{y}}+\frac{1}{\sigma_e+j\omega\varepsilon}\left[\biggl(\frac{\partial^2}{\partial{x^2}}+(k^2=-\gamma^2)\biggr)A_x+\frac{\partial^2{A_y}}{\partial{x}\partial{y}}+\frac{\partial^2{A_z}}{\partial{x}\partial{z}}\right]\label{eqn:Ext}\\
    E_y&=\frac{\partial{F_z}}{\partial{x}}-\frac{\partial{F_x}}{\partial{z}}+\frac{1}{\sigma_e+j\omega\varepsilon}\left[\frac{\partial^2{A_x}}{\partial{x}\partial{y}}+\biggl(\frac{\partial^2}{\partial{y^2}}+(k^2=-\gamma^2)\biggr)A_y+\frac{\partial^2{A_z}}{\partial{y}\partial{z}}\right]\label{eqn:Eyt}\\
    E_z&=\frac{\partial{F_x}}{\partial{y}}-\frac{\partial{F_y}}{\partial{x}}+\frac{1}{\sigma_e+j\omega\varepsilon}\left[\frac{\partial^2{A_x}}{\partial{x}\partial{z}}+\frac{\partial^2{A_y}}{\partial{y}\partial{z}}+\biggl(\frac{\partial^2}{\partial{z^2}}+(k^2=-\gamma^2)\biggr)A_z\right]\label{eqn:Ezt}
\end{align}
\subsubsection{$TM_n$ Solutions}
It is apparent by examining (\ref{eqn:Hxt})-(\ref{eqn:Hzt}) and
(\ref{eqn:Ext})-(\ref{eqn:Ezt}) that in order for a solution to be
$TM_n$ ($H_n=0$), where n is the given direction, all that is
required is that the magnetic vector potential have a component in
that direction with all other components equal to zero, including
the electric vector potential.
\begin{align}
    \vec{A} = A_n\hat{n}&&\vec{F}=0\label{eqn:TMcond}
\end{align}
where $n=x,y,z$. (\ref{eqn:Hxt})-(\ref{eqn:Hzt}) and
(\ref{eqn:Ext})-(\ref{eqn:Ezt}) reduce to they following given the
direction in which they are TM.
\newline
\hrule
$$TM_x$$
\hrule
\begin{align}
    H_x&=0\label{eqn:TMxHx}\\
    H_y&=\frac{\partial A_x}{\partial z}\label{eqn:TMxHy}\\
    H_z&=-\frac{\partial{A_x}}{\partial{y}}\label{eqn:TMxHz}
\end{align}
\begin{align}
    E_x&=\left(\frac{1}{\sigma_e+j\omega\varepsilon}\right)\biggl(\frac{\partial^2}{\partial{x^2}}+(k^2=-\gamma^2)\biggr)A_x\label{eqn:TMxEx}\\
    E_y&=\left(\frac{1}{\sigma_e+j\omega\varepsilon}\right)\frac{\partial^2{A_x}}{\partial{x}\partial{y}}\label{eqn:TMxEy}\\
    E_z&=\left(\frac{1}{\sigma_e+j\omega\varepsilon}\right)\frac{\partial^2{A_x}}{\partial{x}\partial{z}}\label{eqn:TMxEz}
\end{align}
\hrule
$$TM_y$$
\hrule
\begin{align}
    H_x&=-\frac{\partial{A_y}}{\partial{z}}\label{eqn:TMyHx}\\
    H_y&=0\label{eqn:TMyHy}\\
    H_z&=\frac{\partial A_y}{\partial x}\label{eqn:TMyHz}
\end{align}
\begin{align}
    E_x&=\left(\frac{1}{\sigma_e+j\omega\varepsilon}\right)\frac{\partial^2{A_y}}{\partial{x}\partial{y}}\label{eqn:TMyEx}\\
    E_y&=\left(\frac{1}{\sigma_e+j\omega\varepsilon}\right)\biggl(\frac{\partial^2}{\partial{y^2}}+(k^2=-\gamma^2)\biggr)A_y\label{eqn:TMyEy}\\
    E_z&=\left(\frac{1}{\sigma_e+j\omega\varepsilon}\right)\frac{\partial^2{A_y}}{\partial{y}\partial{z}}\label{eqn:TMyEz}
\end{align}
\hrule
$$TM_z$$
\hrule
\begin{align}
    H_x&=\frac{\partial A_z}{\partial y}\label{eqn:TMzHx}\\
    H_y&=-\frac{\partial{A_z}}{\partial{x}}\label{eqn:TMzHy}\\
    H_z&=0\label{eqn:TMzHz}
\end{align}
\begin{align}
    E_x&=\left(\frac{1}{\sigma_e+j\omega\varepsilon}\right)\frac{\partial^2{A_z}}{\partial{x}\partial{z}}\label{eqn:TMzEx}\\
    E_y&=\left(\frac{1}{\sigma_e+j\omega\varepsilon}\right)\frac{\partial^2{A_z}}{\partial{y}\partial{z}}\label{eqn:TMzEy}\\
    E_z&=\left(\frac{1}{\sigma_e+j\omega\varepsilon}\right)\biggl(\frac{\partial^2}{\partial{z^2}}+(k^2=-\gamma^2)\biggr)A_z\label{eqn:TMzEz}
\end{align}
\subsubsection{$TE_n$ Solutions}
It is apparent by examining (\ref{eqn:Hxt})-(\ref{eqn:Hzt}) and
(\ref{eqn:Ext})-(\ref{eqn:Ezt}) that in order for a solution to be
$TE_n$ ($E_n=0$), where n is the given direction, all that is
required is that the electric vector potential have a component in
that direction with all other components equal to zero, including
the magnetic vector potential.
\begin{align}
    \vec{A}=0&&\vec{F}=F_n\hat{n}&&\label{eqn:TEcond}
\end{align}
where $n=x,y,z$. (\ref{eqn:Hxt})-(\ref{eqn:Hzt}) and
(\ref{eqn:Ext})-(\ref{eqn:Ezt}) reduce to they following given the
direction in which they are TE.
\hrule
$$TE_x$$
\hrule
\begin{align}
    H_x&=\left(\frac{1}{\sigma_m+j\omega\mu}\right)\biggl(\frac{\partial^2}{\partial{x^2}}+(k^2=-\gamma^2)\biggr)F_x\label{eqn:THxEx}\\
    H_y&=\left(\frac{1}{\sigma_m+j\omega\mu}\right)\frac{\partial^2{F_x}}{\partial{x}\partial{y}}\label{eqn:THxEy}\\
    H_z&=\left(\frac{1}{\sigma_m+j\omega\mu}\right)\frac{\partial^2{F_x}}{\partial{x}\partial{z}}\label{eqn:THxEz}
\end{align}
\begin{align}
    E_x&=0\label{eqn:TExEx}\\
    E_y&=-\frac{\partial F_x}{\partial z}\label{eqn:TExEy}\\
    E_z&=\frac{\partial{F_x}}{\partial{y}}\label{eqn:TExEz}
\end{align}
\hrule
$$TE_y$$
\hrule
\begin{align}
    H_x&=\left(\frac{1}{\sigma_m+j\omega\mu}\right)\frac{\partial^2{F_y}}{\partial{x}\partial{y}}\label{eqn:TEyHx}\\
    H_y&=\left(\frac{1}{\sigma_m+j\omega\mu}\right)\biggl(\frac{\partial^2}{\partial{y^2}}+(k^2=-\gamma^2)\biggr)F_y\label{eqn:TEyHy}\\
    H_z&=\left(\frac{1}{\sigma_m+j\omega\mu}\right)\frac{\partial^2{F_y}}{\partial{y}\partial{z}}\label{eqn:TEyHz}
\end{align}
\begin{align}
    E_x&=\frac{\partial{F_y}}{\partial{z}}\label{eqn:TEyEx}\\
    E_y&=0\label{eqn:TEyEy}\\
    E_z&=-\frac{\partial F_y}{\partial x}\label{eqn:TEyEz}
\end{align}
\hrule
$$TE_z$$
\hrule
\begin{align}
    H_x&=\left(\frac{1}{\sigma_m+j\omega\mu}\right)\frac{\partial^2{F_z}}{\partial{x}\partial{z}}\label{eqn:TEzHx}\\
    H_y&=\left(\frac{1}{\sigma_m+j\omega\mu}\right)\frac{\partial^2{F_z}}{\partial{y}\partial{z}}\label{eqn:TEzHy}\\
    H_z&=\left(\frac{1}{\sigma_m+j\omega\mu}\right)\biggl(\frac{\partial^2}{\partial{z^2}}+(k^2=-\gamma^2)\biggr)F_z\label{eqn:TEzHz}
\end{align}
\begin{align}
    E_x&=-\frac{\partial F_z}{\partial y}\label{eqn:TEzEx}\\
    E_y&=\frac{\partial{F_z}}{\partial{x}}\label{eqn:TEzEy}\\
    E_z&=0\label{eqn:TEzEz}
\end{align}

Substituting the separated variable (\ref{eqn:phi}) into
(\ref{eqn:TMxHx})--(\ref{eqn:TEzEz}) reduces the field
reconstruction equations to the following. \hrule
$$TM_x$$
\hrule
\begin{align}
    H_x&=0\label{eqn:TMxHxSep}\\
    H_y&=XYZ^\prime\label{eqn:TMxHySep}\\
    H_z&=-X{Y^\prime}Z\label{eqn:TMxHzSep}
\end{align}
\begin{align}
    E_x&=\frac{1}{\sigma_e+j\omega\varepsilon}(k_\perp^2=-\gamma_\perp^2)XYZ\label{eqn:TMxExSep}\\
    E_y&=\frac{1}{\sigma_e+j\omega\varepsilon}X'Y'Z\label{eqn:TMxEySep}\\
    E_z&=\frac{1}{\sigma_e+j\omega\varepsilon}X'YZ'\label{eqn:TMxEzSep}
\end{align}
where,
$k_\perp^2=k_y^2+k_z^2=-\gamma_\perp^2=-\gamma_y^2-\gamma_z^2$
\hrule
$$TM_y$$
\hrule
\begin{align}
    H_x&=-XYZ'\label{eqn:TMyHxSep}\\
    H_y&=0\label{eqn:TMyHySep}\\
    H_z&=X'YZ\label{eqn:TMyHzSep}
\end{align}
\begin{align}
    E_x&=\frac{1}{\sigma_e+j\omega\varepsilon}X'Y'Z\label{eqn:TMyExSep}\\
    E_y&=\frac{1}{\sigma_e+j\omega\varepsilon}(k_\perp^2=-\gamma_\perp^2)XYZ\label{eqn:TMyEySep}\\
    E_z&=\frac{1}{\sigma_e+j\omega\varepsilon}XY'Z'\label{eqn:TMyEzSep}
\end{align}
where,
$k_\perp^2=k_x^2+k_z^2=-\gamma_\perp^2=-\gamma_x^2-\gamma_z^2$
\hrule
$$TM_z$$
\hrule
\begin{align}
    H_x&=XY'Z\label{eqn:TMzHxSep}\\
    H_y&=-X'YZ\label{eqn:TMzHySep}\\
    H_z&=0\label{eqn:TMzHzSep}
\end{align}
\begin{align}
    E_x&=\frac{1}{\sigma_e+j\omega\varepsilon}X'YZ'\label{eqn:TMzExSep}\\
    E_y&=\frac{1}{\sigma_e+j\omega\varepsilon}XY'Z'\label{eqn:TMzEySep}\\
    E_z&=\frac{1}{\sigma_e+j\omega\varepsilon}(k_\perp^2=-\gamma_\perp^2)XYZ\label{eqn:TMzEzSep}
\end{align}
where,
$k_\perp^2=k_x^2+k_y^2=-\gamma_\perp^2=-\gamma_x^2-\gamma_y^2$
\hrule
$$TE_x$$
\hrule
\begin{align}
    H_x&=\frac{1}{\sigma_m+j\omega\mu}(k_\perp^2=-\gamma_\perp^2)XYZ\label{eqn:TExHxSep}\\
    H_y&=\frac{1}{\sigma_m+j\omega\mu}X'Y'Z\label{eqn:TExHySep}\\
    H_z&=\frac{1}{\sigma_m+j\omega\mu}X'YZ'\label{eqn:TExHzSep}
\end{align}
\begin{align}
    E_x&=0\label{eqn:TExExSep}\\
    E_y&=-XYZ'\label{eqn:TExEySep}\\
    E_z&=XY'Z\label{eqn:TExEzSep}
\end{align}
where,
$k_\perp^2=k_y^2+k_z^2=-\gamma_\perp^2=-\gamma_y^2-\gamma_z^2$
\hrule
$$TE_y$$
\hrule
\begin{align}
    H_x&=\frac{1}{\sigma_m+j\omega\mu}X'Y'Z\label{eqn:TEyHxSep}\\
    H_y&=\frac{1}{\sigma_m+j\omega\mu}(k_\perp^2=-\gamma_\perp^2)XYZ\label{eqn:TEyHySep}\\
    H_z&=\frac{1}{\sigma_m+j\omega\mu}XY'Z'\label{eqn:TEyHzSep}
\end{align}
\begin{align}
    E_x&=XYZ'\label{eqn:TEyExSep}\\
    E_y&=0\label{eqn:TEyEySep}\\
    E_z&=-X'YZ\label{eqn:TEyEzSep}
\end{align}
where,
$k_\perp^2=k_x^2+k_z^2=-\gamma_\perp^2=-\gamma_x^2-\gamma_z^2$
\hrule
$$TE_z$$
\hrule
\begin{align}
    H_x&=\frac{1}{\sigma_m+j\omega\mu}X'YZ'\label{eqn:TEzHxSep}\\
    H_y&=\frac{1}{\sigma_m+j\omega\mu}XY'Z'\label{eqn:TEzHySep}\\
    H_z&=\frac{1}{\sigma_m+j\omega\mu}(k_\perp^2=-\gamma_\perp^2)XYZ\label{eqn:TEzHzSep}
\end{align}
\begin{align}
    E_x&=-XY'Z\label{eqn:TEzExSep}\\
    E_y&=X'YZ\label{eqn:TEzEySep}\\
    E_z&=0\label{eqn:TEzEzSep}
\end{align}
where,
$k_\perp^2=k_x^2+k_y^2=-\gamma_\perp^2=-\gamma_x^2-\gamma_y^2$
